The development of mixed and hybrid finite element formulations with the aim of ensuring reliable, robust and accurate formulations has been an ongoing area of research over the past decades. The present work is intended to contribute to this and deals with the application and analysis of mixed and hybrid formulations for nonlinear material relations. The focus here is on the consideration of mixed stress-displacement formulations and the application for hyperelasticity and plasticity.
In particular, the least-squares finite element method (LSFEM) is considered. Among other advantages, the LSFEM results in a minimization problem compared to other mixed methods and thus offers the possibility of direct application of the method in the field of hyperelasticity and finite plasticity without any constraint by stability conditions. For a preliminary investigation of mixed formulations and their fulfillment of plastic material constraints, a Hellinger-Reissner formulation for small deformations is first analyzed. The LSFEM and its challenges are investigated for the small and finite deformation theory and the findings are then considered in the area of finite plasticity. Furthermore, a mixed hybrid finite element formulation based on a hyperelastic least-squares approach is derived and analyzed. For this purpose, continuity requirements are relaxed and enforced by the method of Lagrange multipliers on the inter-element boundaries.